Octahedron orbitals

Symmetry of orbitals on the B6 octahedron, (a) Six outward-pointing (sp) orbitals used for a bonding to 6 H. (b) Six inward-pointing (sp) orbitals used to form the fi framework bonding molecular orbital, (c) ComponenLS for one of the fiM framework bonding molecular orbitals — the other two molecular orbitals are in the yz and zr planes, (d) (Components ftM one of the t2 framework bonding molecular orbitals — the other two molecular orbitals are in the yz and x planes. 

We need six orbitals to accommodate six electron pairs around an atom in an octahedral arrangement, as in SF6 and XeF4, and so we need to use two d-orbitals in addition to the valence s- and p-orbitals to form six sp3d2 hybrid orbitals (Fig. 3.18). These identical orbitals point toward the six corners of a regular octahedron. 

For elements adjacent to the noble gases the principal orbitals used in bond formation are those formed by hybridisation of the s and p orbitals. For the transition elements there are nine stable orbitals to be taken into consideration, which in general are hybrids of five d orbitals, one s orbital, and three p orbitals. An especially important set of six bond orbitals, directed toward the comers of a regular octahedron, are the d2sps orbitals, which are involved in most of the Werner octahedral complexes formed by the transition elements. 

Another way to view the barycentre rule is to consider first the bringing up to the metal of a spherical shell of negative charge which increases the energies of all five d orbitals equally. Then, in this notional picture, if the spherical shell of charge redistributes towards the apices of an octahedron, those orbitals directed towards those apices suffer a further repulsion and energy increase, while those directed in between, acquire a relative stability. 

The left superscript indicates that the arrangements are all spin triplets. The letter T refers to the three-fold degeneracy just discussed and it is in upper case because the symbol pertains to a many-electron (here two) wavefunction (we use lower-case letters for one-electron wavefunctions or orbitals, remember). The subscript g means the wavefunctions are even under inversion through the centre of symmetry possessed by the octahedron (since each d orbital is of g symmetry, so also is any product of them), and the right subscript 1 describes other symmetry properties we need not discuss here. More will be said about such term symbols in the next two sections. 

We shall shortly draw on both of these symmetry and energy aspects of Fig. 6-1 in the construction of molecular orbitals for the octahedron. First, however, let us extend the picture to molecules with more than two atoms. 

The structure of CaB contains bonding bands typical of the boron sublattice and capable of accommodating 20 electrons per CaB formula, and separated from antibonding bands by a relatively narrow gap (from 1.5 to 4.4 eV) . The B atoms of the B(, octahedron yield only 18 electrons thus a transfer of two electrons from the metal to the boron sublattice is necessary to stabilize the crystalline framework. The semiconducting properties of M B phases (M = Ca, Sr ", Ba, Eu, Yb ) and the metallic ones of M B or M B5 phases (Y, La, Ce, Pr, Nd ", Gd , Tb , Dy and Th ) are directly explained by this model . The validity of these models may be questionable because of the existence and stability of Na,Ba, Bft solid solutions and of KB, since they prove that the CaB -type structure is still stable when the electron contribution of the inserted atom is less than two . A detailed description of physical properties of hexaborides involves not only the bonding and antibonding B bands, but also bonds originating in the atomic orbitals of the inserted metal . ... 

Scheme 5 Orbital energy levels of the regular octahedron...

For the triply degenerate orbitals t2g and flM only one of each is plotted for each of them, two further, equal orbitals exist which are oriented along the other two octahedron axes. Right energy sequence of the seven occupied bonding orbitals... 

KT1 does not have the NaTl structure because the K+ ions are too large to fit into the interstices of the diamond-like Tl- framework. It is a cluster compound K6T16 with distorted octahedral Tig- ions. A Tig- ion could be formulated as an electron precise octahedral cluster, with 24 skeleton electrons and four 2c2e bonds per octahedron vertex. The thallium atoms then would have no lone electron pairs, the outside of the octahedron would have nearly no valence electron density, and there would be no reason for the distortion of the octahedron. Taken as a closo cluster with one lone electron pair per T1 atom, it should have two more electrons. If we assume bonding as in the B6Hg- ion (Fig. 13.11), but occupy the t2g orbitals with only four instead of six electrons, we can understand the observed compression of the octahedra as a Jahn-Teller distortion. Clusters of this kind, that have less electrons than expected according to the Wade rules, are known with gallium, indium and thallium. They are called hypoelectronic clusters their skeleton electron numbers often are 2n or 2n — 4. 

FIGURE 5.16 Crystal-field pictures for the low-spin d5 system in deformed octahedra. (a) relative energy of d-orbitals for an octahedron with strong crystal field (b) d-orbitals of the T2s set in a rhombically distorted octahedron (c) three possible electron configurations of increasing total energy (d) the term scheme for the configurations in (c). 

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